Note that \(\mathinner {|{\chi _0}\rangle }\)  is
the canonical purification of a maximally mixed state \(\rho _{AB} = 1_{AB}/(\chi _A \chi _B)\)  - this is the density matrix \(\rho _{AB}\)  that usually arises from a Haar random state reduced to \(AB\)  when \(\chi _C > \chi _A \chi _B\) .
And, \(\mathinner {|{\chi _1}\rangle }\)  is the natural guess for the canonical purification of the original Haar random state reduced to \(AB\)  when \(\chi _C < \chi _A \chi _B\) : that is, when the entanglement wedge is connected.
The doubling procedure is motivated by the AdS/CFT discussion in [1]}.
